出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2013/04/15 22:33:39」(JST)
中性脂肪（ちゅうせいしぼう、neutral fat）ないし中性脂質（ちゅうせいししつ、neutral lipid）とは、脂肪酸のグリセリンエステルを指す。狭義には常温で固体の中性脂質を中性脂肪と呼ぶ[1]。
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詳細は「グリセリン脂肪酸エステル」を参照
グリセリン脂肪酸エステルにはモノグリセリド、ジグリセリド、トリグリセリドが存在するが、血液中に含まれる中性脂肪のほとんどはトリグリセリドである。したがって、中性脂肪はトリグリセリドと同義とする場合も多い。TG、TAGまたはTrigという略号で記されることが多い。脂肪酸とグリセリンが結びついて中性を示すので「中性脂肪」と言う。
中性脂肪の成分である脂肪酸は動物においてはステアリン酸、パルミチン酸など飽和脂肪酸が主であるのに対し植物においてはオレイン酸、リノール酸、リノレン酸のような不飽和脂肪酸を多く含む。したがって、動物性の中性脂肪は室温で固体であるものが多いのに対して、植物性の中性脂肪は室温で液体の場合がほとんどである。
生体内においては、エネルギー貯蔵物質としての役割が大きい。砂漠に生息するラクダや卵殻内での鳥類では中性脂質を酸化して水分に転化する場面もある。また細胞中では部分的に脂肪酸を失った中性脂質（モノグリセリド、ジグリセリド）が細胞内での情報伝達物質として働くことも分かっている。細胞膜は、中性脂肪から取り出された脂肪酸を原料としたリン脂質から形成されている。
生活習慣病における中性脂肪の扱いは複雑で、一時期は完全に無視されるに至ったこともあった。つまりLDLコレステロール（悪玉コレステロール）やHDLコレステロール（善玉コレステロール）が重要とされ、中性脂肪は軽視された。人間の体内の中性脂肪が1000mg/dLを超えると、急性膵炎のリスクが上昇すると考えられており、それさえ抑えればよいと考えられた。
現在世界の高脂血症治療の最先端・最高峰を示すATP-IIIというステートメントでは中性脂肪も立派に補正すべき物質へと戻った。特にメタボリックシンドロームの診断基準に取り入れられ注目されている。
また、肝硬変や肝臓がんなどの原因となるC型肝炎ウイルス（HCV）は、細胞内の中性脂肪を利用して増殖しており、さらに、ウイルスの「コア」と呼ばれるたんぱく質の働きで、細胞内の中性脂肪が増加すると報告され、治療に応用されることが期待されている[2][3]。
中性脂肪についての血液検査の参考基準値は以下のとおりである。
| 項目 | 被験者のタイプ | 下限値 | 上限値 | 単位 | 最適範囲 |
| 中性脂肪 | 10 – 39 歳 | 54[4] | 110[4] | mg/dL | < 100 mg/dL[5] or 1.1[5] mmol/L |
| 0.61[6] | 1.2 [6] | mmol/L | |||
| 40 – 59 歳 | 70[4] | 150[4] | mg/dL | ||
| 0.77[6] | 1.7[6] | mmol/L | |||
| > 60 歳 | 80[4] | 150[4] | mg/dL | ||
| 0.9[6] | 1.7[6] | mmol/L |
| [ヘルプ] |
| Trigonometry |
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| History Usage |
| Reference |
| Identities Exact constants |
| Laws and theorems |
| Law of sines Law of cosines |
| Calculus |
| Trigonometric substitution Integrals of functions |
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Trigonometry (from Greek trigōnon "triangle" + metron "measure"[1]) is a branch of mathematics that studies triangles and the relationships between the lengths of their sides and the angles between those sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves. The field evolved during the third century BC as a branch of geometry used extensively for astronomical studies.[2] It is also the foundation of the practical art of surveying.
Trigonometry basics are often taught in school either as a separate course or as part of a precalculus course. The trigonometric functions are pervasive in parts of pure mathematics and applied mathematics such as Fourier analysis and the wave equation, which are in turn essential to many branches of science and technology. Spherical trigonometry studies triangles on spheres, surfaces of constant positive curvature, in elliptic geometry. It is fundamental to astronomy and navigation. Trigonometry on surfaces of negative curvature is part of Hyperbolic geometry.
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Sumerian astronomers introduced angle measure, using a division of circles into 360 degrees.[4] They and their successors the Babylonians studied the ratios of the sides of similar triangles and discovered some properties of these ratios, but did not turn that into a systematic method for finding sides and angles of triangles. The ancient Nubians used a similar method.[5] The ancient Greeks transformed trigonometry into an ordered science.[6]
Classical Greek mathematicians (such as Euclid and Archimedes) studied the properties of chords and inscribed angles in circles, and proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. Claudius Ptolemy expanded upon Hipparchus' Chords in a Circle in his Almagest.[7] The modern sine function was first defined in the Surya Siddhanta, and its properties were further documented by the 5th century Indian mathematician and astronomer Aryabhata.[8] These Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, and were applying them to problems in spherical geometry.[citation needed] At about the same time, Chinese mathematicians developed trigonometry independently, although it was not a major field of study for them. Knowledge of trigonometric functions and methods reached Europe via Latin translations of the works of Persian and Arabic astronomers such as Al Battani and Nasir al-Din al-Tusi.[9] One of the earliest works on trigonometry by a European mathematician is De Triangulis by the 15th century German mathematician Regiomontanus. Trigonometry was still so little known in 16th century Europe that Nicolaus Copernicus devoted two chapters of De revolutionibus orbium coelestium to explaining its basic concepts.
Driven by the demands of navigation and the growing need for accurate maps of large areas, trigonometry grew into a major branch of mathematics.[10] Bartholomaeus Pitiscus was the first to use the word, publishing his Trigonometria in 1595.[11] Gemma Frisius described for the first time the method of triangulation still used today in surveying. It was Leonhard Euler who fully incorporated complex numbers into trigonometry. The works of James Gregory in the 17th century and Colin Maclaurin in the 18th century were influential in the development of trigonometric series.[12] Also in the 18th century, Brook Taylor defined the general Taylor series.[13]
If one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a triangle is completely determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, regardless of the overall size of the triangle. If the length of one of the sides is known, the other two are determined. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:
The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle, and one of the two sides adjacent to angle A. The adjacent leg is the other side that is adjacent to angle A. The opposite side is the side that is opposite to angle A. The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively. Many English speakers find it easy to remember what sides of the right triangle are equal to sine, cosine, or tangent, by memorizing the word SOH-CAH-TOA (see below under Mnemonics).
The reciprocals of these functions are named the cosecant (csc or cosec), secant (sec), and cotangent (cot), respectively:
The inverse functions are called the arcsine, arccosine, and arctangent, respectively. There are arithmetic relations between these functions, which are known as trigonometric identities. The cosine, cotangent, and cosecant are so named because they are respectively the sine, tangent, and secant of the complementary angle abbreviated to "co-".
With these functions one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines. These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and their included angle or two angles and a side or three sides are known. These laws are useful in all branches of geometry, since every polygon may be described as a finite combination of triangles.
The above definitions apply to angles between 0 and 90 degrees (0 and π/2 radians) only. Using the unit circle, one can extend them to all positive and negative arguments (see trigonometric function). The trigonometric functions are periodic, with a period of 360 degrees or 2π radians. That means their values repeat at those intervals. The tangent and cotangent functions also have a shorter period, of 180 degrees or π radians.
The trigonometric functions can be defined in other ways besides the geometrical definitions above, using tools from calculus and infinite series. With these definitions the trigonometric functions can be defined for complex numbers. The complex exponential function is particularly useful.
See Euler's and De Moivre's formulas.
Graphing process of y = sin(x) using a unit circle.
Graphing process of y = tan(x) using a unit circle.
Graphing process of y = csc(x) using a unit circle.
A common use of mnemonics is to remember facts and relationships in trigonometry. For example, the sine, cosine, and tangent ratios in a right triangle can be remembered by representing them as strings of letters. For instance, a mnemonic for English speakers is SOH-CAH-TOA:
One way to remember the letters is to sound them out phonetically (i.e., SOH-CAH-TOA, which is pronounced 'so-kə-tow'-uh').[14] Another method is to expand the letters into a sentence, such as "Some Old Hippy Caught Another Hippy Trippin' On Acid".[15]
Trigonometric functions were among the earliest uses for mathematical tables. Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy. Slide rules had special scales for trigonometric functions.
Today scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan, and sometimes cis and their inverses. Most allow a choice of angle measurement methods: degrees, radians and, sometimes, grad.[citation needed] Most computer programming languages provide function libraries that include the trigonometric functions. The floating point unit hardware incorporated into the microprocessor chips used in most personal computers have built-in instructions for calculating trigonometric functions.[citation needed]
There are an enormous number of uses of trigonometry and trigonometric functions. For instance, the technique of triangulation is used in astronomy to measure the distance to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. The sine and cosine functions are fundamental to the theory of periodic functions such as those that describe sound and light waves.
Fields that use trigonometry or trigonometric functions include astronomy (especially for locating apparent positions of celestial objects, in which spherical trigonometry is essential) and hence navigation (on the oceans, in aircraft, and in space), music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many physical sciences, land surveying and geodesy, architecture, phonetics, economics, electrical engineering, mechanical engineering, civil engineering, computer graphics, cartography, crystallography and game development.
Identities are those equations that hold true for any value.
Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. Some identities equate an expression to a different expression involving the same angles. These are listed in List of trigonometric identities. Triangle identities that relate the sides and angles of a given triangle are listed below.
In the following identities, A, B and C are the angles of a triangle and a, b and c are the lengths of sides of the triangle opposite the respective angles.
The law of sines (also known as the "sine rule") for an arbitrary triangle states:
where R is the radius of the circumscribed circle of the triangle:
Another law involving sines can be used to calculate the area of a triangle. Given two sides and the angle between the sides, the area of the triangle is:
The law of cosines (known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles:
or equivalently:
The law of tangents:
Euler's formula, which states that , produces the following analytical identities for sine, cosine, and tangent in terms of e and the imaginary unit i:
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